%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\item % 1
\newcommand{\ARA}{
习题1. 用 $ g(x) $ 除 $ f(x) $，求商 $ q(x) $ 与余式 $ r(x) $ :

1) $ f(x) = x^3 - 3x^2 - x - 1 $, $ g(x) = 3x^2 - 2x + 1 $;

2) $ f(x) = x^4 - 2x + 5 $, $ g(x) = x^2 - x + 2 $.

}

\newcommand{\ARAa}{
用 $ f(x)=x^4 - 2x + 5 $ 除以 $ g(x)=x^2 - x + 2$, 求商 $ q(x) $ 与余式 $ r(x) $. 
}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\item % 2
\newcommand{\ARB}{
习题2. $ m, p, q $ 适合什么条件时，有

1) $ x^2 + mx - 1 \mid x^3 + px + q $;

2) $ x^2 + mx + 1 \mid x^4 + px^2 + q $.

}

\newcommand{\ARBa}{
设 $ x^2 + mx + 1$ 整除 $x^4 + px^2 + q$, 求 $m, p, q$ 需要适合的条件。
}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\item % 3
\newcommand{\ARC}{
习题3. 求 $ g(x) $ 除 $ f(x) $ 的商 $ q(x) $ 与余式 $ r(x) $ :

1) $ f(x) = 2x^5 - 5x^3 - 8x $, $ g(x) = x + 3 $;

2) $ f(x) = x^3 - x^2 - x $, $ g(x) = x - 1 + 2i $.

}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\item % 4
\newcommand{\ARD}{
习题4. 把 $ f(x) $ 表成 $ x - x_0 $ 的方幂和，即表成 $ c_0 + c_1 (x - x_0) + c_2 (x - x_0)^2 + \cdots $ 的形式：

1) $ f(x) = x^5 $, $ x_0 = 1 $;

2) $ f(x) = x^4 - 2x^2 + 3 $, $ x_0 = -2 $;

3) $ f(x) = x^4 + 2ix^3 - (1+i)x^2 - 3x + 7 + i $, $ x_0 = -i $.

}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\item % 5
\newcommand{\ARE}{
习题5. 求 $ f(x) $ 与 $ g(x) $ 的最大公因式：

1) $ f(x) = x^4 + x^3 - 3x^2 - 4x - 1 $, $ g(x) = x^3 + x^2 - x - 1 $;

2) $ f(x) = x^4 - 4x^3 + 1 $, $ g(x) = x^3 - 3x^2 + 1 $;

3) $ f(x) = x^4 - 10x^2 + 1 $, $ g(x) = x^4 - 4\sqrt{2}x^3 + 6x^2 + 4\sqrt{2}x + 1 $.

}

\newcommand{\AREa}{
使用辗转相除法，求 $ f(x)=x^4 + x^3 - 3x^2 - 4x - 1 $ 与 $ g(x)=x^3 + x^2 - x - 1 $ 的最大公因式。

}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\item % 6
\newcommand{\ARF}{
习题6. 求 $ u(x), v(x) $，使 $ u(x)f(x) + v(x)g(x) = (f(x), g(x)) $ :

1) $ f(x) = x^4 + 2x^3 - x^2 - 4x - 2 $, $ g(x) = x^4 + x^3 - x^2 - 2x - 2 $;

2) $ f(x) = 4x^4 - 2x^3 - 16x^2 + 5x + 9 $, $ g(x) = 2x^3 - x^2 - 5x + 4 $;

3) $ f(x) = x^4 - x^3 - 4x^2 + 4x + 1 $, $ g(x) = x^2 - x - 1 $.

}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\item % 7
\newcommand{\ARG}{
习题7. 设 $ f(x) = x^3 + (1+t)x^2 + 2x + 2u $, $ g(x) = x^3 + tx + u $ 的最大公因式是一个二次多项式，求 $ t, u $ 的值.

}

\newcommand{\ARGa}{
设 $f(x) = x^3 + (1+t)x^2 + 2x + 2u$ 与 $g(x) = x^3 + tx + u$ 的最大公因式是一个二次多项式，求 $ t, u $ 的值.

}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\item % 8
\newcommand{\ARH}{
习题8. 证明：如果 $ d(x) \mid f(x) $，$ d(x) \mid g(x) $，且 $ d(x) $ 为 $ f(x) $ 与 $ g(x) $ 的一个组合，那么 $ d(x) $ 是 $ f(x) $ 与 $ g(x) $ 的一个最大公因式.

}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\item % 9
\newcommand{\ARI}{
习题9. 证明：$ (f(x)h(x), g(x)h(x)) = (f(x), g(x))h(x) $ （$ h(x) $ 的首项系数为 1）.

}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\item % 10
\newcommand{\ARJ}{
习题10. 如果 $ f(x), g(x) $ 不全为零，证明：
$$
\left(\frac{f(x)}{(f(x), g(x))}, \frac{g(x)}{(f(x), g(x))}\right) = 1.
$$

}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\item % 11
\newcommand{\ARK}{
习题11. 证明：如果 $ f(x), g(x) $ 不全为零，且
$$
u(x)f(x) + v(x)g(x) = (f(x), g(x)),
$$
那么 $ (u(x), v(x)) = 1 $.

}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\item % 12
\newcommand{\ARL}{
习题12. 证明：如果 $ (f(x), g(x)) = 1 $，$ (f(x), h(x)) = 1 $，那么
$$
(f(x), g(x)h(x)) = 1.
$$

}

\newcommand{\ARLa}{
证明：如果 $(f(x), g(x)) = 1$, $(f(x), h(x)) = 1$, 那么 $(f(x), g(x)h(x)) = 1$. 

}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\item % 13
\newcommand{\ARM}{
习题13. 设 $ f_1(x), \ldots, f_m(x), g_1(x), \ldots, g_n(x) $ 都是多项式，而且
$$
(f_i(x), g_j(x)) = 1 \quad (i=1,2,\ldots,m; j=1,2,\ldots,n),
$$
求证：$ (f_1(x)f_2(x)\cdots f_m(x), g_1(x)g_2(x)\cdots g_n(x)) = 1 $.

}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\item % 14
\newcommand{\ARN}{
习题14. 证明：如果 $ (f(x), g(x)) = 1 $，那么 $ (f(x)g(x), f(x)+g(x)) = 1 $.
}

\newcommand{\ARNa}{
证明：如果 $ (f(x), g(x)) = 1 $, 那么 $ (f(x)g(x), f(x)+g(x)) = 1 $.
}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\item % 15
\newcommand{\ARO}{
习题15. 求多项式的公共根：
$$
f(x) = x^3 + 2x^2 + 2x + 1, \quad g(x) = x^4 + x^3 + 2x^2 + x + 1. 
$$
}

\newcommand{\AROa}{
求多项式 $f(x) = x^3 + 2x^2 + 2x + 1$ 与 $g(x) = x^4 + x^3 + 2x^2 + x + 1$ 的公共根。
}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\item % 16
\newcommand{\ARP}{
习题16. 判别下列多项式有无重因式:

1) $ f(x) = x^5 - 5x^4 + 7x^3 - 2x^2 + 4x - 8 $;

2) $ f(x) = x^4 + 4x^2 - 4x - 3 $.

}

\newcommand{\ARPa}{
判别多项式$ f(x) = x^4 + 4x^2 - 4x - 3 $ 有无重因式。
}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\item % 17
\newcommand{\ARQ}{
习题17. 求 $ t $ 值使 $ f(x) = x^3 - 3x^2 + tx - 1 $ 有重根.

}

\newcommand{\ARQa}{
求 $ t $ 值使 $ f(x) = x^3 - 3x^2 + tx - 1 $ 有重根。

}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\item % 18
\newcommand{\ARR}{
习题18. 求多项式 $ x^3 + px + q $ 有重根的条件.

}

\newcommand{\ARRa}{
求多项式 $ x^3 + px + q $ 有重根的条件。
}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\item % 19
\newcommand{\ARS}{
习题19. 如果 $ (x-1)^2 \mid Ax^4 + Bx^2 + 1 $，求 $ A, B $.

}

\newcommand{\ARSa}{
如果 $ (x-1)^2 \mid Ax^4 + Bx^2 + 1 $，求 $ A, B $ 的值。

}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\item % 20
\newcommand{\ART}{
习题20. 证明：$ 1 + x + \frac{x^2}{2!} + \cdots + \frac{x^n}{n!} $ 不能有重根.

}

\newcommand{\ARTa}{
证明多项式 $ 1 + x + \frac{x^2}{2!} + \cdots + \frac{x^n}{n!} $ 没有重根。

}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\item % 21
\newcommand{\ARU}{
习题21. 如果 $ a $ 是 $ f'''(x) $ 的一个 $ k $ 重根，证明：$ a $ 是

$$
g(x) = \frac{x-a}{2}[f'(x) + f'(a)] - f(x) + f(a)
$$

的一个 $ k+3 $ 重根.

}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\item % 22
\newcommand{\ARV}{
习题22. 证明：$ x_0 $ 是 $ f(x) $ 的 $ k $ 重根的充分必要条件是 $ f(x_0) = f'(x_0) = \cdots = f^{(k-1)}(x_0) = 0 $，而 $ f^{(k)}(x_0) \neq 0 $.

}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\item % 23
\newcommand{\ARW}{
习题23. 举例说明断语“如果 $ \alpha $ 是 $ f'(x) $ 的 $ m $ 重根，那么 $ \alpha $ 是 $ f(x) $ 的 $ m+1 $ 重根”是不对的.

}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\item % 24
\newcommand{\ARX}{
习题24. 证明：如果 $ (x-1) \mid f(x^n) $，那么 $ (x^n-1) \mid f(x^n) $.

}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\item % 25
\newcommand{\ARY}{
习题25. 证明：如果 $ (x^2+x+1) \mid f_1(x^3) + xf_2(x^3) $，那么

$$
(x-1) \mid f_1(x), \quad (x-1) \mid f_2(x).
$$

}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\item % 26
\newcommand{\ARZ}{
习题26. 将多项式 $ x^n - 1 $ 在复数范围内和在实数范围内因式分解.

}

\newcommand{\ARZa}{
将多项式 $ x^5 - 1 $ 在复数范围内和在实数范围内因式分解。
}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\item % 27
\newcommand{\ARZA}{
习题27. 求下列多项式的有理根：

1) $ x^3 - 6x^2 + 15x - 14 $;

2) $ 4x^4 - 7x^2 - 5x - 1 $;

3) $ x^5 + x^4 - 6x^3 - 14x^2 - 11x - 3 $.

}

\newcommand{\ARZAa}{
求多项式 $ x^3 - 6x^2 + 15x - 14 $ 的有理根。
}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\item % 28
\newcommand{\ARZB}{
习题28. 判断下列多项式在有理数域上是否可约：

1) $ x^2 + 1 $;

2) $ x^4 - 8x^3 + 12x^2 + 2 $;

3) $ x^6 + x^3 + 1 $;

4) $ x^p + px + 1 $，$ p $ 为奇素数；

5) $ x^4 + 4kx + 1 $，$ k $ 为整数.

}

\newcommand{\ARZBa}{
判断多项式 $ x^4 - 8x^3 + 12x^2 + 2 $ 在有理数域上是否可约。
}







